CS 70
Discrete Mathematics and Probability Theory
Spring 2010
Alistair Sinclair
Note 19
A Brief Introduction to Continuous Probability
Up to now we have focused exclusively on
discrete
probability spaces
Ω
, where the number of sample points
ω
∈
Ω
is either Fnite or countably inFnite (such as the integers). As a consequence we have only been able
to talk about
discrete
random variables, which take on only a Fnite (or countably inFnite) number of values.
But in real life many quantities that we wish to model probabilistically are
real-valued
; examples include
the position of a particle in a box, the time at which an certain incident happens, or the direction of travel
of a meteorite. In this lecture, we discuss how to extend the concepts we’ve seen in the discrete setting to
this continuous setting. As we shall see, everything translates in a natural way once we have set up the right
framework. The framework involves some elementary calculus but (at this level of discussion) nothing too
scary.
Continuous uniform probability spaces
Suppose we spin a “wheel of fortune” and record the position of the pointer on the outer circumference of
the wheel. Assuming that the circumference is of length
°
and that the wheel is unbiased, the position is
presumably equally likely to take on any value in the real interval
[
0
,°
]
. How do we model this experiment
using a probability space?
Consider for a moment the (almost) analogous discrete setting, where the pointer can stop only at a Fnite
number
m
of positions distributed evenly around the wheel. (If
m
is very large, then presumably this is in
some sense similar to the continuous setting.) Then we would model this situation using the discrete sample
space
Ω
=
{
0
,
°
m
,
2
°
m
,...,
(
m
−
1
)
°
m
}
, with uniform probabilities Pr
[
ω
]=
1
m
for each
ω
∈
Ω
. In the continuous
world, however, we get into trouble if we try the same approach. If we let
ω
range over all real numbers
in
[
0
,°
]
, what value should we assign to each Pr
[
ω
]
? By uniformity this probability should be the same for
all
ω
, but then if we assign to it any positive value, the sum of all probabilities Pr
[
ω
]
for
ω
∈
Ω
will be
∞
!
Thus Pr
[
ω
]
must be zero for all
ω
∈
Ω
. But if all of our sample points have probability zero, then we are
unable to assign meaningful probabilities to any events!
To rescue this situation, consider instead any non-empty
interval
[
a
,
b
]
⊆
[
0
,°
]
. Can we assign a non-zero
probability value to this interval? Since the total probability assigned to
[
0
,°
]
must be 1, and since we want
our probability to be uniform, the logical value for the probability of interval
[
a
,
b
]
is
length of
[
a
,
b
]
length of
[
0
,°
]
=
b
−
a
°
.
In other words, the probability of an interval is proportional to its length.
Note that intervals are subsets of the sample space
Ω
and are therefore
events
. So in continuous probability,
we are assigning probabilities to certain basic events, in contrast to discrete probability, where we assigned
probability to
points
in the sample space. But what about probabilities of other events? Actually, by speci-